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Yang-Baxter sigma models, proposed by Klimcik and Delduc-Magro-Vicedo, havebeen recognized as a powerful framework for studying integrable deformations oftwo-dimensional non-linear sigma models. In this short article, as an importantgeneralization, we review a non-integrable sigma model in the Yang-Baxter sigmamodel approach based on [arXiv:1406.2249]. In particular, we discuss a familyof deformations of the 5D Sasaki-Einstein manifold $T^{1,1}$, instead of thestandard deformations of the $5$-sphere S$^5$. For this purpose, we firstdescribe a novel construction of $T^{1,1}$ as a supercoset, and provide aphysical interpretation of this construction from viewpoint of the dualKlebanov-Witten field theory. Secondly, we consider a $3$-parameter deformationof $T^{1,1}$ by using classical $r$-matrices satisfying the classicalYang--Baxter equation (CYBE). The resulting metric and NS-NS two-formcompletely agree with the ones previously obtained via TsT (T-dual -- shift --T-dual) transformations, and contain the Lunin-Maldacena background as aspecial case. Our result indicates that what we refer to as thegravity/CYBE(Classical Yang-Baxter Equation) correspondence can be extendedbeyond integrable cosets.

Towards the gravity/CYBE correspondence beyond integrability -- Yang-Baxter deformations of $T^{1,1}$